H_rref_phi_ladder
The declaration defines the hypothesis that a reference length r_ref exists on the phi-ladder, specifically r_ref = ell0 * phi^N for integer N, such that the gravitational running ratio G_ratio at 20 nm is within 1 of 32. Researchers modeling nanoscale corrections to Newton's constant would cite this to fix the scale where effective G reaches 32 times its macroscopic value. The definition is a direct existential statement over the phi-ladder and the G_ratio function.
claimThere exists an integer $N$ and positive real $r_{ref} = ell_0 phi^N$ such that $|G_{ratio}(20 nm, r_{ref}) - 32| < 1$.
background
ell0 is the fundamental length unit in RS-native units, defined as 1 in the voxel scale and equivalently c * tau0 where tau0 derives from hbar G / (pi c^5). The phi-ladder refers to discrete scales ell0 * phi^N for integer N, with phi the golden-ratio fixed point forced by the self-similar solution of the Recognition Composition Law. G_ratio(r, r_ref) encodes the running of Newton's constant as G_eff(r) = G_infty * (1 + |beta| * (r / r_ref)^beta) with beta = -(phi-1)/phi^5. The module sets the local context as gravitational running at nanometer scales, where G(r) strengthens from its macroscopic limit G_infty toward an enhancement factor of 32 at r approximately 20 nm.
proof idea
One-line definition that directly states the existential claim over N in Z and r_ref in R, using the ell0 constant and the G_ratio function already imported from Constants and the local module.
why it matters in Recognition Science
This hypothesis anchors the reference scale r_ref inside the phi-ladder for the running-G model, directly supporting the module's prediction that G_eff reaches approximately 32 G_infty at 20 nm. It closes the link between the T6 phi fixed point and the nanoscale gravity correction without introducing new axioms. No downstream theorems are recorded yet, leaving the hypothesis available for later discharge by an explicit construction of N.
scope and limits
- Does not prove existence of any such N or r_ref.
- Does not derive the numerical value 32 or the exponent beta.
- Does not connect the hypothesis to laboratory data or Casimir-force measurements.
- Does not specify the functional form of G_ratio beyond its use in the existential statement.
formal statement (Lean)
175def H_rref_phi_ladder : Prop :=
proof body
Definition body.
176 ∃ N : ℤ, ∃ r_ref : ℝ, r_ref = ell0 * phi ^ N ∧ r_ref > 0 ∧
177 abs (G_ratio 20e-9 r_ref - 32) < 1
178
179/-! ## Q10: Casimir Force Correction
180
181Running G at 20nm gives G_eff ≈ 32 * G_inf. But G_inf ≈ 6.7e-11 makes
182even the enhanced gravitational force negligible vs Casimir (~10 Pa at 20nm).
183The fractional gravitational correction to Casimir is ≈ 2e-18. -/
184
185/-- Gravitational pressure between two plates. -/